Teacher
Guide
Multiplication
& Division
Created By
Marilyn Burns
About the Program
 Introduction to Do The Math Now!
Program Overview From Marilyn Burns . . . . . . . . . . . . . . . . . . iv
Proven Instructional Strategies . . . . . . . . . . . . . . . . . . . . . . . . vi
Program Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
 The Lessons
UNIT
1
Build a Foundation for Multiplication . . . . . . . . . . 1–67
Lessons 1–5
Understanding the Meaning of Multiplication . . . . . . . . . . . . . . . . . . . . . . 1
Lessons 6–10
Practicing Multiplication Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Lessons 11–15
Connecting Multiplication to Arrays, Rectangles,
and the Multiplication Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Measure Student Understanding With the
End-of-Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
UNIT
2
Develop Multiplication Number Sense . . . . . 69–135
Lessons 1–5
Identifying Patterns on the Multiplication Chart . . . . . . . . . . . . . . . . . 69
Lessons 6–10
Using the Strategy of “Splitting” to Multiply . . . . . . . . . . . . . . . . . . . . . 91
Lessons 11–15
Learning to Multiply by Multiples of 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Measure Student Understanding With the
End-of-Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
UNIT
3
Use Place-Value Strategies to Multiply . . . . 137–201
Lessons 1–5
Using the Distributive Property to Multiply . . . . . . . . . . . . . . . . . . . . . . 137
Lessons 6–10
Making Estimates and Figuring Products . . . . . . . . . . . . . . . . . . . . . . . . 157
Lessons 11–15
Solving Multi-Digit Multiplication Problems . . . . . . . . . . . . . . . . . . . . . 179
Measure Student Understanding With the
End-of-Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
ii
Introduction
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UNIT
4
Connect Multiplication and Division . . . . . . 203–269
Lessons 1–5
Solving Division Grouping Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Lessons 6–10
Understanding Divisibility by 2, 5, and 10 . . . . . . . . . . . . . . . . . . . . . . 225
Lessons 11–15
Calculating Quotients and Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Measure Student Understanding With the
End-of-Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
UNIT
5
Use Place-Value Strategies to Divide . . . . . . 271–337
Lessons 1–5
Solving Division Problems in Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Lessons 6–10
Solving Division Problems With Three-Digit Dividends . . . . . . . . . 293
Lessons 11–15
Extending Division to Two-Digit Divisors . . . . . . . . . . . . . . . . . . . . . . . . 315
Measure Student Understanding With the
End-of-Unit Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
 Additional Resources
Objectives Tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Beginning-of-Unit and End-of-Unit Assessments . . . . . . . . . . . . . . . . . . . 341
Assessment Answer Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Do The Math Community News . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .366
Teacher Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Teacher Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385
iii
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Program Overview From Marilyn Burns
Dear Colleague,
Serena and Gabe are examples of struggling math students I’ve met
during my almost 50 years of teaching.
I showed Serena, a sixth grader, a division problem with four choices for
the answer.
425  4
50
100
200
400
Marilyn: Which of these numbers do you think is closest to the answer?
(I point to the four choices.)
Thanks to the
Do The Math Team
Serena: (Thinks for a moment and then asks.) Can I use paper and pencil?
Eunice Hendrix-Martin, teacher
San Marcos School District, Carlsbad, CA
Marilyn: Try and figure it out in your head.
Leo Kostelnik, teacher
Bolinas-Stinson Union School District, Bolinas, CA
Kris Lee, teacher
Park Day School, Oakland, CA
Danielle Ross, teacher
Mill Valley School District, Mill Valley, CA
Mallika Scott, teacher
Lighthouse Community Charter School, Oakland, CA
Maryann Wickett, teacher
San Marcos School District, Carlsbad, CA
Lynne Zolli, teacher (retired)
San Francisco School District, San Francisco, CA
Barbara Bobb, teacher (retired) and editor
Sandwich Public Schools, Sandwich, IL
Thanks to the
Do The Math Schools
Serena: (Thinks again and then muses.) I know the number has to be
smaller than 400 because division makes things smaller.
(Then she lowers her head and begins to “write” with her finger
on the desk, setting up the problem as long division to work out
the answer.)
Sadly, there are far too many students like Serena in our math classes,
who have learned computational procedures but have not developed the
ability to reason numerically in other ways.
I gave Gabe, a seventh grader identified by his teacher as a struggling
math student, a fraction addition problem.
_1
Our thanks to the following school districts, teachers,
administrators, and their students for contributing to
a summer pilot study or photography.
New York City Schools, NY
Linda Curtis-Bey, Lisa Edmond, Alex Spencer,
Robert O’Leary, Julie Matthews, Kimberly Tan,
Mary Lou Wainwright
P.S. 20, The Anna Silver School, New York, NY
Pasadena Unified, CA
Dr. Jacqueline Cochran, Joan Morris, Sandra Mims,
Jose Guzman, Julie Young, Dennis Silverio,
Elsa Aguirre de Tonay, Jeff Bracamonte
Englewood Public School District, NJ
2
Marilyn: You don’t have to figure out the exact answer to this problem.
I’m interested in whether you think the answer is greater than
1 or less than 1.
Gabe:
(Looks at the problem carefully and then responds.) It’s less
than 1.
Marilyn: How do you know?
Gabe:
iv
2
_
5
I added across the tops and across the bottoms and got threesevenths, and I know that three-sevenths is less than one.
Introduction
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Gabe made one of the most common fraction errors, following a faulty
procedure instead of thinking about the numbers at hand. As stated in
the Common Core State Standards for Mathematics, “Students who lack
understanding of a topic may rely on procedures too heavily. Without a
flexible base from which to work, they may be less likely to . . . deviate
from a known procedure to find a shortcut.” This lack of understanding
prevents Serena, Gabe, and other students like them from developing
needed reasoning skills.
I was a middle school math teacher for the first eight years of my teaching
career. In all my classes, there were always some students who were
woefully ill prepared. They usually had some skill with paper-and-pencil
computation, but had learned these procedures by rote and would quickly
become lost when presented with a situation that differed even slightly
from exactly what they were used to seeing. Math rarely made sense to
them. In fact, they didn’t even expect math to make sense. Their goal was
to “do the page,” not to “do the math.” They were rarely asked to explain
their reasoning, and when they were, they were unable to do so.
We created Do The Math Now! to meet the needs of the thousands of
middle and high school students who, like Serena and Gabe, need to
develop essential math understanding and skills. We decided that
the best support we could provide these students would be to focus
on multiplication, division, and fractions—topics that are critical
foundations for the students’ continued math success with algebra.
Marilyn Burns is one of
today’s most highly respected
and trusted mathematics
educators. She is the founder of
Math Solutions, an organization
dedicated to the improvement
of math instruction in our
nation’s schools. Over the course
of almost 50 years, Marilyn
has worked with students and
teachers in classrooms across
the country.
Marilyn’s experiences have given
her a unique insight into how
to help students overcome the
stumbling blocks that prevent
them from being successful with
mathematics. In collaboration
with Scholastic, Marilyn and a
team of Math Solutions master
classroom teachers developed
Do The Math, an intervention
program that provides teachers
with the tools and support they
need to help students turn these
stumbling blocks into building
blocks of mathematical success.
So here it is, a yearlong course for middle and high school students
who need math support in addition to their regular math classes.
Do The Math Now! is filled with the same kinds of scaffolded and paced
lessons, games, and activities that have been the mainstay of the success
of Do The Math.
I’m pleased to present this support for finally building a foundation
of essential math understanding and skills that all students need
and deserve.
v
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unit
2
contents
Lessons 1–5
Identifying patterns on the
multiplication chart . . . . . . . 69–90
Lessons 1–15
Exploring patterns in products . . . . . . . . .72
In this unit, students are actively engaged with
the Commutative Property of Multiplication,
the Associative Property of Multiplication,
and the Distributive Property of Multiplication
over Addition. Relying on the visual model
of rectangles, students further develop their
understanding and number sense.
Exploring patterns in multiples of
4, 5, and 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Identifying facts to practice . . . . . . . . . . 84
Assessing student understanding . . . . . 88
Lessons 6–10
Using the strategy of
“splitting”to multiply . . . . 91–112
Using rectangles to figure products . . . 94
Using splitting to multiply by 11 . . . . . . 98
Using splitting to multiply by 12 . . . . . 102
Using the number-splitting strategy
to multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Assessing student understanding . . . . 110
Lessons 11–15
Learning to multiply by
multiples of 10 . . . . . . . . . . . . 113–134
Students will…
Figuring products for
three factors . . . . . . . . . . . . . . . . . . . . . . . . . 116
• Represent rectangles with multiplication equations.
• Calculate products with factors 0–12.
• Multiply two-digit factors by 10.
Multiplying using the
“times 10” rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120
• Use the Associative and Commutative Properties of Multiplication to
Multiplying by multiples of 10 . . . . . . . 124
• Communicate ideas with key math vocabulary: Associative Property
Playing Target 300,
a multiplication game . . . . . . . . . . . . . . . 128
calculate products.
of Multiplication, Commutative Property of Multiplication, equation,
factor, multiple, multiplication, and product.
≈
1
2
3
3
1
2
3
4
5
6
7
4
5
6
4
5
6
7
8
9
10
11
12
7
8
9
10
11
12
2
22 24
1
16 18 20
14
12
8 10
2 4 6
30 33 36
21 24 27
18
15
12
44 48
3 6 9
32 36 40
8
2
4
2
0
16 2
4 8 12
50 55 60
35 40 45
0
3
5
2
0
2
66 72
5 10 15
48 54 60
2
4
6
3
0
24 3
84
6 12 18
63 70 77
42 49 56
5
3
8
2
7 14 21
80 88 96
56 64 72
8
4
0
4
4 32
Assessing student understanding . . . . 132
Measuring Student
Understanding With the
End-of-Unit Assessment . . . . . . . . . . 135
unit 2
Develop
Multiplication
Number Sense
Exploring patterns on a
multiplication chart . . . . . . . . . . . . . . . . . . . 76
FROM MaRilyn BuRns
Dear Colleague,
These lessons introduce students to the strategy of “rectangle splitting” to figure
products, a tool that informally introduces the Distributive Property of Multiplication
over Addition.
35
UNIT
2
Lessons
6–10
14
14
While students already know that the order of factors does not affect the products—
that 4 3 3 and 3 3 4, for example, have the same product—here they are formally
introduced to the Commutative Property of Multiplication.
3 ≈ 4 = 12
4 ≈ 3 = 12
To practice the splitting strategy, students multiply by 11. Because they are familiar
with the pattern of the products from 11 3 1 to 11 3 9 and already know the answers,
students are able to focus on applying the strategy.
11
10
1
80
11
≈8
≈ 8 = 80
≈8=8
+ 8 = 88
≈ 8 = 88
Using the Strategy of
“Splitting” to Multiply
35 + 14 + 14 = 63
Then students tackle problems of multiplying by 12. The lessons help them move from
splitting rectangles to splitting numbers, and then to figuring products mentally.
DTM_MAD_TG_U2_091.indd 91
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PLANNER
Lesson 6
Lesson 7
Lesson Summary
Students find products greater than
36 by drawing rectangles and figuring
the number of squares in them.
Students use the number-splitting
strategy to multiply by 11.
Objectives
•Introduce key math vocabulary:
•Represent rectangles with equations.
Commutative Property of Multiplication.
•Represent rectangles with equations.
•Calculate products with factors 0
through 12.
•Use the Commutative Property of
•Calculate products with factors
0 through 12.
•Communicate ideas with key math
vocabulary: equation, multiplication,
product, and times.
Multiplication to solve problems.
•Communicate ideas with key math
vocabulary: Commutative Property of
Multiplication, equation, factor,
multiplication, product, and times.
Materials
 Teacher Bag
S
 Student Bag
c
 Chart
Built-in
Differentiation
• WorkSpace pages 54, 55, 190
and 195–198
• Cut-Out Rectangles c
• Grid Chart c
• Math Vocabulary chart
• scissors
• WorkSpace pages 56 and 57
• Grid Chart c
• Unit 2: Do The Math Community News
Clear visual directions in the WorkSpace
provide support for students who are
not yet proficient with written or verbal
directions.
Recording each equation on the board when
demonstrating rectangle splitting provides
visual support before the added challenge
of figuring products mentally.
Interactive Whiteboard Tools
contains all hands-on manipulatives and
WorkSpace pages for Unit 2, Lessons 6–10.
92
Develop Multiplication
Number Sense
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Lesson 8
Lesson 9
Lesson 10
Students use the number-splitting
strategy to multiply by 12.
Students figure products mentally
by using the number-splitting
strategy to multiply the factors
11 and 12.
Students demonstrate understanding
of the objectives of Lessons 6–9
by completing WorkSpace pages
independently.
•Represent rectangles with equations.
•Recall products for factors
•Recall products for factors
•Calculate products with factors
0 through 12.
•Communicate ideas with key
math vocabulary: equation, factor,
multiplication, product, and times.
0 through 12.
•Calculate products with factors
0 through 12.
•Communicate ideas with key
math vocabulary: equation, factor,
multiplication, product, and times.
Assess
0 through 12.
•Calculate products with factors
0 through 12.
•Communicate ideas with key math
vocabulary: equation, multiplication, and
product.
• WorkSpace page 60
• blank paper
• WorkSpace pages 61–67
• Pathways Game Board A
• green tiles
• Additional Practice
The carefully paced transition from
the geometric model of rectangle to
the numeric model of number splitting
supports students transitioning to the
goal of figuring products mentally.
Communication with partners as
they use the number-splitting strategy
reinforces the concept and the
language used to express it.
Assessing students with familiar
problems allows students to show their
understanding without having to approach
the material in an unfamiliar format.
2
Lessons
6–10
Using the Strategy of
“Splitting” to Multiply
• WorkSpace pages 58 and 59
• Multiplication Chart c
• Grid Chart c
UNIT
TeacherSpace CD-ROM
contains Unit Assessments, Additional Practice
pages, and other reproducibles to support teaching
these lessons.
DTM_MAD_TG_U2_092-093.indd 93
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7 Using splitting to multiply by 11
Lesson 27
WhoLe Group
Lesson Summary
Students use the number-splitting strategy to
multiply by 11.
Step
Objectives
•Represent rectangles with equations.
•Calculate products with factors 0 through 12.
•Communicate ideas with key math vocabulary:
equation, multiplication, product, and times.
Materials
1
Introduce the lesson.
h Today, we’ll practice multiplying by 11 using a strategy that
I think will help you figure the products more easily.
2
C  Chart
•WorkSpace pages 56 and 57
•Grid Chart C
•Unit 2: Do The Math Community News
1
Students split a rectangle to figure 11 3 8.
h Look at WorkSpace page 56. The first multiplication
problem is 11 times 8. Let’s say the answer together. (88)
The rectangle on the page matches the problem 11 times 8
because it has 11 rows with 8 squares in each row.
Now show how you might figure the product by rectangle
splitting. You may talk with your partners about how to
do this, but each of you should show your work on your
own page.
Interactive Whiteboard Tools
WorkSpace pages and manipulatives for Lesson 7 are
provided on the Interactive Whiteboard Tools CD-ROM.
Preparation
Unit 2: Do The Math Community News
Make 1 copy for each student from page 367 or the
TeacherSpace CD-ROM.
Review figuring products
by rectangle splitting.
Give students time to solve the problem. Then call on
students to present their solutions. Students may split
the rectangle in different ways; acknowledge all of their
approaches.
Workspace page 56
Rectangle Splitting: 11
Language Development
directions
Key Math Vocabulary
 Draw a rectangle to match the multiplication problem.
 Split the rectangle to make two easier problems.
SPANISH
equation
ecuación
multiplication
multiplicación
product
producto
times
por
 Write equations for your split rectangle.
unit 2
ENGLISH
 Write the product in the box for each equation.
11  8  88
11  9  99
11  13  143
80
90
Academic Vocabulary
ENGLISH
SPANISH
rectangle
rectángulo
row
fila
strategy
estrategia
9
8
80  8  88
90  9  99
130
13
130  13  143
56
UNIT 2
■
Lesson 7
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98
6/14/11 10:47:53 AM
Develop Multiplication Number Sense
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Last Lesson Students find products
greater than 36 by drawing rectangles and
figuring the numbers of squares in them.
Lesson 7 Students use the
number-splitting strategy to multiply by 11.
Next Lesson Students use the
number-splitting strategy to multiply by 12.
Whole Group
2
Step
3
Model splitting the rectangle into
10  8 and 1  8.
Tell students that you want to show how to split
the rectangle so that you can multiply by 10.
Explain that multiplying by 10 is easy and that
splitting a rectangle this way will help solve
problems with greater numbers.
h Here’s how to split an 11-by-8 rectangle so that
one of the new rectangles has 10 rows.
Draw the rectangle on the Grid Chart and then
draw a horizontal line as shown.
1
Guide students as they
relate rectangles and
equations.
Demonstrate writing equations for an
11  8 rectangle.
h Why do I want one rectangle to be 10 by 8?
Have students talk with their partners about the
advantage of the 10 3 8 rectangle. Choose a
student to explain. (To multiply by 10, we can tack
on a zero.)
Now make the connection between the rectangles
and their equations. Write 10 3 8 5 80 and
1 3 8 5 8 as shown.
10 ≈ 8 = 80
1≈8=8
Explain that you changed the original problem
into two easier problems, just as you split the
rectangle into two smaller rectangles. Instead
of solving 11 3 8, you solved 10 3 8 and 1 3 8.
Then you can add the answers.
Write on the board as shown.
Do The Math
Distribute the copies of the News.
This News provides directions for a
multiplication game which gives students
practice multiplying numbers from 7 to 12.
11
10
1
80
11
≈8
≈8
≈8
+8
≈8
= 80
= 8
= 88
= 88
Continue
UNIT 2
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■
Lesson 7
99
6/23/11 12:36:29 PM
continued
lesson 7 Using splitting to multiply by 11
whole group
3
Step
1
Demonstrate the
number-splitting
strategy.
Demonstrate how to solve 11  9 by
splitting the factor 11.
h Now let’s try another problem. We’ll solve it just
by thinking about the numbers, without drawing
a rectangle.
Write 11 3 9 on the board. Ask the students to say
the answer together.
SUPPORTING INSTRUCTION
Explain to students that although they may already
know the product of 11  9, you’re using the problem
as a way to learn a multiplication strategy. After
students learn the strategy, they’ll try it on problems
for which they don’t already know the answer.
h If we drew a rectangle to solve 11 times 9, we
would split it so that instead of one rectangle with
11 rows, we would have a rectangle with 10 rows
and a rectangle with 1 row. We can do the same
thing with just the numbers. Instead of splitting
a rectangle, we split the number 11 into 10  1.
Watch and listen as I do this.
As you explain, record the steps on the board.
h Split 11 so that it’s 10 and 1.
Then multiply each part by 9.
Then add 90 and 9 to get 99.
So 11  9 5 99.
11 ≈ 9
11 = 10 + 1
10 ≈ 9 = 90
1≈9=9
90 + 9 = 99
11 ≈ 9 = 99
2
Model solving 11  13 by splitting
the factor 11.
Write 11 3 13 on the board.
h I’ll show you the number-splitting strategy again,
this time to multiply 11 times 13.
Record each step on the board as you explain.
h I can split the 11 into 10 and 1. Then I can
multiply 10 times 13, and 1 times 13. That gives
me 130 plus 13. What is 130 plus 13? (143)
So, 11  13 5 143.
11 ≈ 9
11 = 10 + 1
10 ≈ 9 = 90
1≈9=9
90 + 9 = 99
11 ≈ 9 = 99
11 ≈ 13
11 = 10 + 1
10 ≈ 13 = 130
1 ≈ 13 = 13
130 + 13 = 143
11 ≈ 13 = 143
CONNECTING TO ALGEBRA
Splitting numbers and multiplying to get partial
products develops a conceptual understanding of
the Distributive Property. We don’t introduce the
name of the property; rather, our focus is on making
sense of it—understanding how and why it works
through rectangle and number splitting. With this
foundation, students are less likely to make mistakes
in algebra like multiplying 3(a + 8) and getting
3a + 8. Using and understanding the Distributive
Property is essential for success in algebra.
100 Develop Multiplication Number Sense
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IndIvIdualS
whole group
4
Step
1
Guide students as
they use the numbersplitting strategy.
Model solving 11  15.
h What two multiplication problems would you
5
Students solve
problems
independently.
Step
1
write to solve 11 times 15? (10 3 15 and 1 3 15)
Students use the strategy.
h Now you’ll try two problems on your own.
Have students turn to WorkSpace page 57.
Explain the directions and have students
complete the page.
What is 10 times 15? (150)
What is 1 times 15? (15)
What is 150 plus 15? (165)
Workspace page 57
Record each step on the board.
Number Splitting: 11
Write the product to complete the equation:
11 3 15 5 165.
Directions
11  12
1
2
Split a factor and
write two equations.
11≈15
10≈15=150
1≈15=15
150+15=165
11≈15=165
2
1
11≈12
10≈12=120
1≈12=12
120+12=132
11≈12=132
2
11  17
Equation 1
10  17  170
Equation 2
1  14  14
Equation 2
1  17  17
Write the product.
Add the products.
11  14  154
170  17  187
Write the product.
11  17  187
Split the rectangles below to show how you split the numbers in 1 and 2.
Students may split the second factor; they will find the sum of
11  10 and 11  4, and the sum of 11  10 and 11  7.
UNIT 2
problems would you write to solve 11 times 12?
Write the product to complete the equation:
11 3 12 5 132.
Write the product.
10  14  140
140  14  154
DTM_MAD_WS_U02_L07_056-057.indd 57
(10 3 12 and 1 3 12)
What is 11 times 12? (132)
11  12  132
Add the products of
the two equations.
Equation 1
Add the products.
Model solving 11  12.
h Let’s try another. What two multiplication
What is the product of each problem? (120 and 12)
11  14
3
120 + 12 = 132
unit 2
11 = 10 + 1
10 ≈ 12 = 120
1 ≈ 12 = 12
2
■
Lesson 7
57
6/14/11 10:47:57 AM
Students check their solutions.
Have students compare their solutions with those
of their partners. If two partners have different
solutions, have each one explain to the other how
he or she solved the problem. Have both students
determine which solution is correct.
sUpporTINg INsTrUcTIoN
If a student can’t explain his or her solution, ask
questions such as: How did you split 11? How did
you decide which two equations to use? Where did
you get the numbers that add to 154 (or 187)?
Stop
UNIT 2
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■
Lesson 7
101
6/23/11 12:36:42 PM
10 assessing student understanding
lesson 30
whole group
Lesson Summary
Students demonstrate understanding of the
objectives of Lessons 6–9 by completing
WorkSpace pages independently.
Step
Objectives
•Recall products for factors 0 through 12.
•Calculate products with factors 0 through 12.
•Communicate ideas with key math vocabulary:
equation, multiplication, and product.
1
•WorkSpace pages 61–67
•Pathways Game Board A
•green tiles
•Additional Practice
 Student Bag
 Student Bag
Interactive Whiteboard Tools
WorkSpace pages and manipulatives for Lesson 10 are
provided on the Interactive Whiteboard Tools CD-ROM.
Preparation
Print out a full-page copy of the Pathways Game
Board A from the TeacherSpace CD-ROM.
Adhere it to the board in order to demonstrate the
game to the whole group. Alternatively, you can
demonstrate the game to your students using the
Interactive Whiteboard Tools.
Language Development
Key Math Vocabulary
English
spanish
equation
ecuación
multiplication
multiplicación
product
producto
Academic Vocabulary
English
spanish
strategy
estrategia
Introduce the lesson.
h Today you’ll show what you know by completing WorkSpace
pages independently. Then you’ll play a game called
Pathways that will give you practice with multiplying.
2
Materials
1
Teach a multiplication
game.
Explain how to play Pathways.
Display a Pathways game board.
h The goal is to be the first player to make a continuous
pathway across the game board—top to bottom or side to
side. Each square is a stepping stone.
To show different paths to win, trace your finger across the
board. Start at a top square such as 24 and trace going up
and down or along diagonals (or both) to get to the bottom
of the board (for example: 24, 40, 56, and 49). Then start
at
thePathways
left Game
sideGame
such
as
A30 and
A trace your finger across until
Pathways
Board
Board
you get to the right side (for example: 30, 25, 56, 64, 20).
Explain
long
18 18
32that
24as 24
15 as
48they
32
15
48make a continuous path, they
don’t need to make it in order.
28 28
40 40
35 35
64 64
20 20
3 Demonstrate
30 30
12 12
56 56
21a turn.
16 16
21
Explain that the numbers in each of the squares are
9 9
25 and
49 the
42numbers
36 36
25
49
42
products
beneath are factors.
I choose 2 factors. I’ll choose 3 and 6. I place green
h
3 First
4
5 6
7 8
3 4
5 6
7 8
tiles on the two numbers. 3  6 is 18 so I will mark an X on
18. Then it is my partner’s turn.
A
A
Pathways
GameGame
Board
Pathways
Board
18 18
32 32
24 24
15 15
48 48
28 28
40 40
35 35
64 64
20 20
30 30
12 12
56 56
21 21
16 16
9
25 25
49 49
42 42
36 36
9
3
4
3
5
4
6
5
7
6
8
7
8
Write 3 3 6  18 on the board.
Pathways Game Board A
h Both players write the equation on the game recording
sheets
in the24
WorkSpace.
18 32
15 48
110
Develop Multiplication Number Sense
DTM_MAD_TG_U2_L10_110-111.indd 110
28 40 35 64 20
30
12
56
9
25 49 42 36
3
4
5
21
6
7
16
8
6/23/11 12:43:09 PM
Last Lesson Students figure products
mentally by using the number-splitting
strategy.
Next Lesson Students use properties to
figure products for three factors.
Lesson 10 Students demonstrate
understanding of the objectives of
Lessons 6–9.
individuals
2
Step
4
Demonstrate the other player’s turn.
the first turn, a player can only move one
h After
Pathways Game Board A
green tile to a new number. My partner could
move the green tile off of the 3 and on to any
18 number
32 24or move
15 the
48green tile off of 6 and
other
on to any other number. Let’s say that my partner
28 40
35from
64the 20
moves
the tile
6 to the 7. Now he or
she multiplies 3 times 7 and marks an O on that
30 12 56 21 16
product.
3 3 49
7? (21)
9 is25
42
h What
36
Draw a ring around 21 on the game board.
3 4 5 6 7 8
h We both write the equation on our game
1
Students complete an
assessment.
Students complete WorkSpace pages
61 and 62.
Have students turn to WorkSpace pages 61 and
62. Explain that they should complete both
pages independently and should not use the
multiplication chart to write or verify products.
h When you finish these two pages, you will play
Pathways. The game boards are on WorkSpace
page 64 and you can record your and your
partner’s equations on page 65. If you need
to refer to the rules, they are on WorkSpace
page 63.
recording sheets.
Workspace page 61
Pathways Game Board A
32 24
15
Show What You Know
Show What You Know
directions
directions
48
unit
18
Workspace page 62
28 40 35 64 20
12
56
9
25 49 42 36
3
4
5
21
6
7
16
8
h You and your partner take turns moving only one
The first player to make a path from top to bottom
or side to side is the winner.
62
Workspace page 63
Workspace page 64
How to Play
A
unit
of the tiles on each turn and then marking the
product with an X or O. It is okay if on your turn
you move one tile and place it on the same factor
as the other tile. For example, both tiles could
be on 6. You multiply 6 3 6 and mark 36 on the
game board. Also you may not mark a product
that is already taken by your partner.
61
1
A
A
A
unit
30
2
3 ≈ 6 = 18
A
A
A
A
3
3 ≈ 6 = 18
4 ≈ 6 = 24
63
64
After the lesson
UNIT 2
DTM_MAD_TG_U2_L10_110-111.indd 111
■
Lesson 10
111
6/28/11 4:37:27 PM
continued
Lesson 10 Assessing student understanding
Check Point
Monitor Progress and Differentiate Instruction
Use the Annotated WorkSpace to assess pages 61 and 62. Although these lessons are carefully paced
and scaffolded, there may be instances when students need additional support or challenges.
For Students Who Need More Support
For Students Ready for a Challenge
•Have the student make flash cards of the
multiplication facts that he or she still needs
to learn. Have the student turn over the flash
cards one at a time. As each card is turned
over, suggest ways to figure out the product
if necessary. Allow the student to use paper
and pencil to record equations. Then have the
student state the product. If the product is
incorrect, help the student find the error in his
or her solution.
•Provide the student with a Pathways game card that
has factors greater than 8 (B, C, D, or E) and have
the student play the game by himself or herself.
The goal is to make a pathway in as few turns
as possible. The student should check his or her
products on the Multiplication Chart.
•Play Pathways one-on-one with a student who
needs additional practice using some of the
more difficult facts, such as those on Game
Boards D and E. By doing this you will have
an opportunity to discuss ways to figure an
unknown product. For example, if the student
does not know 6 3 12, you can suggest that
he or she split the 12 into 10 and 2, and
then think: 6 3 10 is 60, 6 3 2 is 12, and
60 1 12 5 72. You are not only helping the
student with the facts, but also strengthening
his or her understanding of splitting.
•Provide grid paper and challenge the student to
draw a 15-by-7 rectangle. Ask the student to
split the rectangle so that there is a group of 10
(splitting the rectangle into a 10-by-7 and a
5-by-7) and to figure the product (70 1 35 5 105).
Alternatively, guide the student to split the rectangle
into three groups of 5 (three 5-by-7 rectangles, or
35 1 35 1 35 5 105). Have the student make up
his or her own rectangles and use splitting to figure
the products. The student can draw rectangles with
13–19 rows with 2–9 tiles in each row.
additionaL practice
Unit 2
›› Lessons 6–10
Name
Date
Practice Number Splitting
2
3
10 ≈ 2 =20
2≈2=4
20 + 4 = 24
Write the product from
memory if you can.
additionaL practice
All students could benefit from additional practice. For your
convenience, the reproducible of the Additional Practice for Lessons
6–10 is available on the TeacherSpace CD-ROM. This Additional
Practice gives students more opportunities to use number splitting to
find products of problems with factors of 11 or 12.
112
Show that the product is
correct using number splitting.
12  15  180
10 ≈ 15 = 150
2 ≈ 15 =
=30
150 + 30 = 180
12 ≈ 15 = 180
If you don’t know the product, use
number splitting to figure the product.
1
12  4 
4
11  6 
7
8  12 
2
8  11 
5
11  11 
8
11  12 
3
12  5 
6
11  10 
9
12  12 
TM ® & © Scholastic Inc. All rights reserved.
1
Develop Multiplication Number Sense
DTM_MAD_TG_PM_U2_L10_112.indd 112
6/28/11 4:32:54 PM
End-of-Unit Assessment
Measure Student Understanding
After completing Lesson 15 and differentiating instruction, administer the Unit 2:
End-of-Unit Assessment found on pages 347 and 348 or on the TeacherSpace CD-ROM.
UNIT
2
For Tracking Student Progress
Lessons
1–15
Use the End-of-Unit Assessment Answer Key on page 362 to determine the number
of items students answered correctly for each of the following unit objectives.
Record their results on the Objectives Tracker found on page 339.
Unit objectives are listed in bold. Lesson objectives are listed below with the
corresponding assessment items in parentheses.
multiplication & division
Objectives Tracker (Units 1–2)
 Record the number of items a student answered correctly for each objective
in the boxes below. For more information about these objectives, see the
Measure Student Understanding pages at the end of each unit.
Multiply whole numbers.
B = Beginning-of-Unit Assessment
E = End-of-Unit Assessment
nt n
ame
s
Calculate products with factors 0
through 10. (Items 4–10)
Calculate products with factors of 11 or 12
using the number-splitting strategy.
(Items 11–13 and 21)
Calculate products with two-digit factors
and 10. (Items 14 and 15)
Multiply one-digit numbers by multiples of
10 from 10 to 100. (Items 16 and 17)
objectives
9
10 10
10 10
10 10
10 10
10 10
10 10
10 10
10 10
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
B
2
E
2
B
2
E
2
B
2
E
2
B
2
E
2
B
2
E
2
B
2
E
2
B
2
E
2
B
2
E
unit 2
B
E
9
9
B
E
9
9
B
E
9
9
B
E
9
9
B
E
9
9
B
E
9
9
B
E
9
9
Represent multiplication of
whole numbers with visual
models.
(Items 1–3)
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Multiply whole numbers.
(Items 4–17 and 21)
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
Utilize the properties of
numbers to solve problems.
(Items 18–20 and 22)
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
Communicate ideas with key
math vocabulary.
(Items 21 and 22)
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
TM ® & © Scholastic Inc. All rights reserved.
9
Multiply whole numbers.
(Items 9–18)
Communicate ideas with key
math vocabulary.
(Items 21 and 22)
Use the Associative Property of
Multiplication to determine the product of
three factors. (Items 18–20 and 22)
Multiplication, factor, product.
(Items 21 and 22)
E
Write and solve
multiplication problems.
(Items 19–20 and 21)
Utilize the properties of numbers to solve
problems.
Communicate ideas with key math
vocabulary: Associative Property of
B
Represent multiplication of
whole numbers with visual
models.
(Items 1–8 and 22)
unit 1
339
DTM_MAD_TG_ObjTracker_339-340.indd 339
Measure student progress and understanding by comparing student results on the
Beginning- and End-of-Unit Assessments.
6/28/11 4:09:42 PM
End-of-Unit Assessment
Represent rectangles with multiplication
equations. (Items 1–3)
stud
e
Represent multiplication of whole
numbers with visual models.
For Further Differentiation
If you find a student is still having difficulty with one or more of the objectives,
revisit the For Students Who Need More Support suggestions found on the Lesson 5,
10, and 15 CheckPoint pages. This student should receive individualized support
to master these objectives during the next unit.
DTM_MAD_TG_PM_U2_L15_134-135.indd 135
6/29/11 9:38:31 AM
On the cover, you’ll find a spiral staircase with landings
separating each section of stairs. In medieval times, spiral
staircases were built against the walls of castles in a
clock-wise direction. This design allowed the defending
swordsman to swing his sword freely while coming down
the stairs. There was also no handrail, which made it
easier to knock his attacker off the stairs.
Marilyn Burns
Multiplication & Division
www.scholastic.com
WorkSpace
Created By
Multiplication & Division
Today we use spiral staircases for less defensive
purposes— mostly to get from floor to floor in a building.
Suppose there are 16 stairs between each landing and the
staircase has 10 circular sections of stairs. How many
stairs are there in all?
WorkSpace
The Math on the Cover
Multiplication Chart
≈
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18 20 22 24
3
3
6
9
12
15
18
21 24 27 30 33 36
4
4
8
12
16 20 24 28 32 36 40 44 48
5
5
10
15 20 25 30 35 40 45 50 55 60
6
6
12
18 24 30 36 42 48 54 60 66 72
7
7
14
21 28 35 42 49 56 63 70
8
8
16 24 32 40 48 56 64 72 80 88 96
9
9
18 27 36 45 54 63 72 81 90 99 108
10
10 20 30 40 50 60 70 80 90 100 110 120
11
11
12
12 24 36 48 60 72 84 96 108 120 132 144
77 88 99 110 121 132
TM ® & © Scholastic Inc. All rights reserved.
22 33 44 55 66
77 84
2
UNIT 1
■
Lesson 1
DTM_MAD_WS_U01_L01_002-004.indd 2
6/14/11 11:15:32 AM
Rectangle Splitting: 11
directions
 Draw a rectangle to match the multiplication problem.
 Split the rectangle to make two easier problems.
unit 2
 Write equations for your split rectangle.
 Write the product in the box for each equation.
11  8  88
80  8  88
11  9  99
11  13  143
80
90
8
9
90  9  99
130
13
130  13  143
56
UNIT 2
■
Lesson 7
DTM_MAD_WS_U02_L07_056-057.indd 56
6/14/11 10:47:53 AM
Number Splitting: 11
Directions
11  12
1
2
Split a factor and
write two equations.
1
11  14
3
120 + 12 = 132
11  12  132
Add the products of
the two equations.
Write the product.
2
11  17
Equation 1
10  14  140
Equation 1
10  17  170
Equation 2
1  14  14
Equation 2
1  17  17
Add the products.
Add the products.
140  14  154
Write the product.
unit 2
11 = 10 + 1
10 ≈ 12 = 120
1 ≈ 12 = 12
11  14  154
170  17  187
Write the product.
11  17  187
Split the rectangles below to show how you split the numbers in 1 and 2.
Students may split the second factor; they will find the sum of
11  10 and 11  4, and the sum of 11  10 and 11  7.
UNIT 2
DTM_MAD_WS_U02_L07_056-057.indd 57
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Lesson 7
57
6/14/11 10:47:57 AM
Show What You Know
directions
 Write the products.
12  12  144
5
12  10  120
2
11  7  77
6
12  5  60
3
11  10  110
7
12  11  132
4
11  12  132
8
11  13  143
unit 2
1
 Use the number-splitting strategy to find the product.
Write each equation.
9
12  13
10  13  130
or
12  10  120
2  13  26
12  3  36
130  26  156
120  36  156
12  13  156
12  13  156
UNIT 2
DTM_MAD_WS_U02_L10_061-067.indd 61
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Lesson 10
61
6/14/11 11:06:17 AM